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Infinitely many solutions for nonlinear Schrödinger system with non-symmetric potentials
1. | Department of Mathematics, University of British Columbia, Vancouver, B.C., V6T 1Z2 |
2. | Department of mathematics, East China Normal University, 500 Dong Chuan Road, Shanghai 200241 |
3. | Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Chile |
References:
[1] |
A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations,, \emph{C. R. Acad. Sci. Paris Ser.}, 1342 (2006), 453.
doi: 10.1016/j.crma.2006.01.024. |
[2] |
T. Bartsch, N. Dancer and Z. Q. Wang, A Liouville theorem, a priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system,, \emph{Cal. Var. Partial Differential Equations.}, 37 (2010), 345.
doi: 10.1007/s00526-009-0265-y. |
[3] |
T. Bartsch, Z. Q. Wang and J. Wei, Bound states for a coupled Schrödinger system,, \emph{J. Fixed Point Theory Appl.}, 2 (2007), 353.
doi: 10.1007/s11784-007-0033-6. |
[4] |
J. Y. Byeon and M. Tanaka, Semiclassical standing waves with clustering peaks for nonlinear Schrödinger equations,, \emph{Memoirs of the American Mathematical Society, 229 (2014).
|
[5] |
K. Chow, Periodic solutions for a system of four coupled nonlinear Schrödinger equations,, \emph{Phys. Rev. Lett. A}, 285 (2001), 319.
doi: 10.1016/S0375-9601(01)00369-3. |
[6] |
M. Conti, S. Terracini and G. Verzini, Nehari's problem and competing species systems,, \emph{Ann. Inst. H. Poincar Anal. Non Linaire}, 19 (2002), 871.
doi: 10.1016/S0294-1449(02)00104-X. |
[7] |
N. Dancer, J. C. Wei and T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system,, \emph{Ann. Inst. H. Poincar Anal. Non Linaire}, 27 (2010), 953.
doi: 10.1016/j.anihpc.2010.01.009. |
[8] |
M. del Pino, J. C. Wei and W. Yao, Intermediate reduction methods and infinitely many positive solutions of nonlinear Schrödinger equations with non-symmetric potentials,, \emph{Car. Var. PDE., 53 (2015), 473.
doi: 10.1007/s00526-014-0756-3. |
[9] |
Y. Guo and J. Wei, Infinitely many positive solutions for nonlinear Schrödinger system with non-symmetric first order,, preprint., (). Google Scholar |
[10] |
F. Hioe and T. Salter, Special set and solution of coupled nonlinear Schrödinger equations,, \emph{J. Phys. A: Math. Gen., 35 (2002), 8913.
|
[11] |
T. C. Lin and J. C. Wei, Ground state of $N$ coupled nonlinear Schrödinger equations in $R^n$, $n\leq 3$,, \emph{Comm. Math Phys.}, 255 (2005), 629.
doi: 10.1007/s00220-005-1313-x. |
[12] |
T. C. Lin and J. C. Wei, Solitary and self-similar solutions of two-component system of nonlinear Schrödinger equations,, \emph{Phy. D}, 220 (2006), 99.
doi: 10.1016/j.physd.2006.07.009. |
[13] |
Z. Liu and Z. Q. Wang, Multiple bound states of nonlinear Schrödinger system,, \emph{Comm. math. Phys.}, 282 (2008), 721.
doi: 10.1007/s00220-008-0546-x. |
[14] |
A. Malchiodi, Some new entire solutions of semilinear elliptic equations on $\R^N$,, \emph{Adv. Math.}, 221 (2009), 1843.
|
[15] |
M. Mitchell and M. Segev, Self-trapping of inconherent white light,, \emph{Nature}, 387 (1997), 880.
doi: 10.1038/43136. |
[16] |
M. Musso, F. Pacard and J. Wei, Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation,, \emph{J. Eur. Math. Soc.}, 14 (2012), 1923.
doi: 10.4171/JEMS/351. |
[17] |
B. Noris, H. Tavares, S. Terracini and G. Verzini, Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition,, \emph{Comm. Pure Appl. Math.}, 63 (2010), 267.
|
[18] |
S. J. Peng and Z. Q. Wang, Segregated and synchronized vector solutions for nonlinear Schrödinger systems,, \emph{Arch. Rational. Mech. Anal.}, 208 (2013), 305.
doi: 10.1007/s00205-012-0598-0. |
[19] |
E. Timmermans, Phase seperation of Bose Einstein condensates,, \emph{Phys. Rev. Lett.}, 81 (1998), 5718. Google Scholar |
[20] |
S. Terracini and G. Verzini, Multipulse phase in $k-$mixtures of Bose-Einstein condenstates,, \emph{Arch. Rat. Mech. Anal.}, 194 (2009), 717.
doi: 10.1007/s00205-008-0172-y. |
[21] |
L. Wang, J. Wei and S. Yan, A Neumann problem with critical exponent in nonconvex domains and Lin-Ni's conjecture,, \emph{Trans. Amer. Math. Soc.}, 362 (2010), 4581.
doi: 10.1090/S0002-9947-10-04955-X. |
[22] |
L. Wang, J. Wei and S. Yan, On Lin-Ni's conjecture in convex domains,, \emph{Proc. Lond. Math. Soc.}, 102 (2011), 1099.
doi: 10.1112/plms/pdq051. |
[23] |
L. Wang and C. Zhao, Solutions with clustered bubbles and a boundary layer of an elliptic problem,, \emph{Discrete Contin. Dyn. Syst.}, 34 (2014), 2333.
|
[24] |
J. C. Wei and T. Weth, Nonradial symmetric bound states for system of two coupled Schrödinger equations,, \emph{Rend. Lincei Mat. Appl.}, 18 (2007), 279.
|
[25] |
J. C. Wei and T. Weth, Radial solutions and phase separation in a system of two coupled Schrödinger equations,, \emph{Arch. Rat. Mech. Anal.}, 190 (2008), 83.
doi: 10.1007/s00205-008-0121-9. |
[26] |
J. C. Wei and S. S. Yan, Infinitely many positive solutions for the nonlinear Schrödinger equations in $R^n$,, \emph{Calc. Var. Partial Differential Equations}, 37 (2010), 423.
doi: 10.1007/s00526-009-0270-1. |
[27] |
J. Wei and W. Yao, Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations,, \emph{CPAA}, 11 (2012), 1003.
|
show all references
References:
[1] |
A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations,, \emph{C. R. Acad. Sci. Paris Ser.}, 1342 (2006), 453.
doi: 10.1016/j.crma.2006.01.024. |
[2] |
T. Bartsch, N. Dancer and Z. Q. Wang, A Liouville theorem, a priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system,, \emph{Cal. Var. Partial Differential Equations.}, 37 (2010), 345.
doi: 10.1007/s00526-009-0265-y. |
[3] |
T. Bartsch, Z. Q. Wang and J. Wei, Bound states for a coupled Schrödinger system,, \emph{J. Fixed Point Theory Appl.}, 2 (2007), 353.
doi: 10.1007/s11784-007-0033-6. |
[4] |
J. Y. Byeon and M. Tanaka, Semiclassical standing waves with clustering peaks for nonlinear Schrödinger equations,, \emph{Memoirs of the American Mathematical Society, 229 (2014).
|
[5] |
K. Chow, Periodic solutions for a system of four coupled nonlinear Schrödinger equations,, \emph{Phys. Rev. Lett. A}, 285 (2001), 319.
doi: 10.1016/S0375-9601(01)00369-3. |
[6] |
M. Conti, S. Terracini and G. Verzini, Nehari's problem and competing species systems,, \emph{Ann. Inst. H. Poincar Anal. Non Linaire}, 19 (2002), 871.
doi: 10.1016/S0294-1449(02)00104-X. |
[7] |
N. Dancer, J. C. Wei and T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system,, \emph{Ann. Inst. H. Poincar Anal. Non Linaire}, 27 (2010), 953.
doi: 10.1016/j.anihpc.2010.01.009. |
[8] |
M. del Pino, J. C. Wei and W. Yao, Intermediate reduction methods and infinitely many positive solutions of nonlinear Schrödinger equations with non-symmetric potentials,, \emph{Car. Var. PDE., 53 (2015), 473.
doi: 10.1007/s00526-014-0756-3. |
[9] |
Y. Guo and J. Wei, Infinitely many positive solutions for nonlinear Schrödinger system with non-symmetric first order,, preprint., (). Google Scholar |
[10] |
F. Hioe and T. Salter, Special set and solution of coupled nonlinear Schrödinger equations,, \emph{J. Phys. A: Math. Gen., 35 (2002), 8913.
|
[11] |
T. C. Lin and J. C. Wei, Ground state of $N$ coupled nonlinear Schrödinger equations in $R^n$, $n\leq 3$,, \emph{Comm. Math Phys.}, 255 (2005), 629.
doi: 10.1007/s00220-005-1313-x. |
[12] |
T. C. Lin and J. C. Wei, Solitary and self-similar solutions of two-component system of nonlinear Schrödinger equations,, \emph{Phy. D}, 220 (2006), 99.
doi: 10.1016/j.physd.2006.07.009. |
[13] |
Z. Liu and Z. Q. Wang, Multiple bound states of nonlinear Schrödinger system,, \emph{Comm. math. Phys.}, 282 (2008), 721.
doi: 10.1007/s00220-008-0546-x. |
[14] |
A. Malchiodi, Some new entire solutions of semilinear elliptic equations on $\R^N$,, \emph{Adv. Math.}, 221 (2009), 1843.
|
[15] |
M. Mitchell and M. Segev, Self-trapping of inconherent white light,, \emph{Nature}, 387 (1997), 880.
doi: 10.1038/43136. |
[16] |
M. Musso, F. Pacard and J. Wei, Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation,, \emph{J. Eur. Math. Soc.}, 14 (2012), 1923.
doi: 10.4171/JEMS/351. |
[17] |
B. Noris, H. Tavares, S. Terracini and G. Verzini, Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition,, \emph{Comm. Pure Appl. Math.}, 63 (2010), 267.
|
[18] |
S. J. Peng and Z. Q. Wang, Segregated and synchronized vector solutions for nonlinear Schrödinger systems,, \emph{Arch. Rational. Mech. Anal.}, 208 (2013), 305.
doi: 10.1007/s00205-012-0598-0. |
[19] |
E. Timmermans, Phase seperation of Bose Einstein condensates,, \emph{Phys. Rev. Lett.}, 81 (1998), 5718. Google Scholar |
[20] |
S. Terracini and G. Verzini, Multipulse phase in $k-$mixtures of Bose-Einstein condenstates,, \emph{Arch. Rat. Mech. Anal.}, 194 (2009), 717.
doi: 10.1007/s00205-008-0172-y. |
[21] |
L. Wang, J. Wei and S. Yan, A Neumann problem with critical exponent in nonconvex domains and Lin-Ni's conjecture,, \emph{Trans. Amer. Math. Soc.}, 362 (2010), 4581.
doi: 10.1090/S0002-9947-10-04955-X. |
[22] |
L. Wang, J. Wei and S. Yan, On Lin-Ni's conjecture in convex domains,, \emph{Proc. Lond. Math. Soc.}, 102 (2011), 1099.
doi: 10.1112/plms/pdq051. |
[23] |
L. Wang and C. Zhao, Solutions with clustered bubbles and a boundary layer of an elliptic problem,, \emph{Discrete Contin. Dyn. Syst.}, 34 (2014), 2333.
|
[24] |
J. C. Wei and T. Weth, Nonradial symmetric bound states for system of two coupled Schrödinger equations,, \emph{Rend. Lincei Mat. Appl.}, 18 (2007), 279.
|
[25] |
J. C. Wei and T. Weth, Radial solutions and phase separation in a system of two coupled Schrödinger equations,, \emph{Arch. Rat. Mech. Anal.}, 190 (2008), 83.
doi: 10.1007/s00205-008-0121-9. |
[26] |
J. C. Wei and S. S. Yan, Infinitely many positive solutions for the nonlinear Schrödinger equations in $R^n$,, \emph{Calc. Var. Partial Differential Equations}, 37 (2010), 423.
doi: 10.1007/s00526-009-0270-1. |
[27] |
J. Wei and W. Yao, Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations,, \emph{CPAA}, 11 (2012), 1003.
|
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